determine the quadratic form matix, eigen value, eigen vector, model matrix, normalized model matrix, canonical form, index, nature, signature, rank for 2x3^2 +5x2^2+3x3^2+4x1x2
Answers
Answer:
Characteristic equation and Eigen value:
For any square matrix A, the equation | | where is a scalar is called characteristic
equation. Here the scalar is called Eigen value.
Eigen vector:
If is a square matrix, a non-zero vector is an Eigenvector of if there is a scalar (lambda)
such that
( )
Properties of Eigen values and Eigen vectors:
1. The sum of Eigen values of a matrix is equal to the sum of diagonal elements of
that matrix.
2. The product of Eigen values of a matrix is equal to the determinant of that
matrix.
3. The Eigen values of upper and lower triangular matrices are its diagonal values.
4. If are the Eigen values of a matrix then the Eigen values of
the matrix
(a)
where m is an integer.
(b) are where a and b are real numbers.
5. If are the Eigen vectors of a matrix then the Eigen vectors
of the matrix
, are where m is an integer, a and b are
real numbers.
6. Eigen vectors are non zero vectors.
7. Eigen vectors are not unique.
Two Eigen values of the matrix ( 1x2) are equal to 1 each. Find the Eigen values of
.
Solution:
Let are the Eigen values of matrix
Given
We know that
⇒
Step-by-step explanation:
matrix is equal to the sum of diagonal elements of
that matrix.
2. The product of Eigen values of a matrix is equal to the determinant of that
matrix.
3. The Eigen values of upper and lower triangular matrices are its diagonal values.
4. If are the Eigen values of a matrix then the Eigen values of
the matrix
(a)
where m is an integer.
(b) are where a and b are real numbers.
5. If are the Eigen vectors of a matrix then the Eigen vectors
of the matrix
, are where m is an integer